How often have you seen your students develop mastery of the cross-multiplication algorithm for solving proportions, yet be lost when asked to apply their knowledge in problem solving, explain why the algorithm works, or transfer their understanding to working with linear equations?

Check out the website of the National Council of Teachers of Mathematics for an excellent research-based discussion on the progression of concepts that supports deep understanding of ratio relationships and proportional reasoning.

Here is their summary:

“A proportion is a relationship of equality between two ratios. Research on students’ understanding of ratio and proportion suggests that developing a conceptual understanding of ratio relationships includes the following:

Learning to attend to two quantities simultaneously

Forming a multiplicative comparison of two quantities; for example, comparing a 25-inch rope to a 10-inch rope by understanding that the first rope is 2.5 times as long as the second

Forming a composed unit, such as a 3:2 unit to describe a class that has 3 girls for every 2 boys, and iterating (repeating) and partitioning (breaking into equal-size parts) the unit

Creating a family of equivalent ratios by iterating and partitioning or by using multiplication and division

Studies have found that a strong foundation in proportional reasoning can support students’ understanding of linear functions and graphs, linear equations in the form y = mx and y = mx + b, and measurement situations. Teachers can promote students’ proportional reasoning capacities by balancing skills and concepts and by delaying the instruction of the cross-multiplication algorithm until students have already gained experience with forming ratios and understanding proportions as an equivalence of ratios.”